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A337156 Numbers k such that the k-th triangular number has all its prime factors congruent to 1 mod 4.

%I A337156 #25 Aug 07 2025 15:16:57
%S A337156 1,25,73,145,169,193,289,313,337,409,457,481,577,625,673,697,745,793,
%T A337156 841,865,985,1009,1129,1153,1201,1249,1321,1345,1369,1417,1465,1489,
%U A337156 1513,1537,1585,1657,1681,1753,1801,1873,1993,2017,2041,2137,2257,2305,2329,2377,2425,2473
%N A337156 Numbers k such that the k-th triangular number has all its prime factors congruent to 1 mod 4.
%C A337156 The k-th triangular number t_k is given as t_k = k(k+1)/2. The t_k associated with this sequence form the intersection of A004613 and A000217.
%C A337156 Apart from 1, numbers whose prime factors are all congruent to 1 mod 4 are also known as primitive hypotenuse numbers because they are candidates for the hypotenuse of primitive right triangles.
%C A337156 For t_k to be a primitive hypotenuse number all its divisors must be congruent to 1 mod 4. Therefore k has to be odd and congruent to 1 mod 8.
%H A337156 Frank M Jackson, <a href="/A337156/b337156.txt">Table of n, a(n) for n = 1..10000</a>
%e A337156 a(2) = 25 because the 25th triangular number is 325, the prime factorization of 325 is 5^2*13, and 5,13 are both congruent to 1 mod 4. It is the second such occurrence.
%t A337156 lst={}; Do[p=1+8n;If[Union@Mod[First/@FactorInteger[p(p+1)/2], 4]=={1}, AppendTo[lst, p]], {n, 0, 10^3}]; lst
%t A337156 Select[Range[2500],Union[Mod[FactorInteger[(#(#+1))/2][[;;,1]],4]]=={1}&] (* _Harvey P. Dale_, Aug 07 2025 *)
%o A337156 (PARI) isok(k) = my(f=factor(k*(k+1)/2)[,1]~); #select(x->((x%4)==1), f) == #f; \\ _Michel Marcus_, Nov 22 2020
%Y A337156 Cf. A000217, A004613.
%K A337156 nonn
%O A337156 1,2
%A A337156 _Frank M Jackson_, Nov 21 2020